Metastates in finite-type mean-field models: visibility, invisibility, and random restoration of symmetry

TL;DR

This paper studies disordered mean-field models with finite states, analyzing metastates and phase transitions, revealing conditions for pure states visibility, and showing equal weights for two pure states without symmetry.

Contribution

It introduces a geometric construction to distinguish visible and invisible states and derives explicit weights for pure states in finite-type mean-field models.

Findings

Only pure states are observed under non-degeneracy.

Invisible states have zero weights, visible states have explicit weights.

Two pure states must have equal weights if only two are present.

Abstract

We consider a general class of disordered mean-field models where both the spin variables and disorder variables take finitely many values. To investigate the size-dependence in the phase-transition regime we construct the metastate describing the probabilities to find a large system close to a particular convex combination of the pure infinite-volume states. We show that, under a non-degeneracy assumption, only pure states are seen, with non-random probability weights for which we derive explicit expressions in terms of interactions and distributions of the disorder variables. We provide a geometric construction distinguishing invisible states (having zero weights) from visible ones. As a further consequence we show that, in the case where precisely two pure states are available, these must necessarily occur with the same weight, even if the model has no obvious symmetry relating the two.